OF ARTS AND SCIENCES. 315 



to BS, and AN parallel to CT, respectively intersecting the straight 

 line BC at E and F. 



Upon Lambert's hypothesis, equal amounts of light will emanate 

 from all straight liues at a sufficient distance from the observer, inter- 

 cepted between the parallels AN, BT', and equally illuminated ; that 

 is, so illuminated that equal lengths of different lines receive equal 

 quantities of incident light. This condition will be fulfilled (if the 

 intensity of the illumination remains the same for all the lines) when 

 the liues make equal angles with the rays of light. Suppose the 

 furrowed cylinder to be inscribed in a smooth cylinder, and BC to 

 represent a sensibly straight portion of the circle bounding the in- 

 tersection of the new cylinder with the plane containing the lines 

 AB, AC. At the same time, let the system of lines directed to the 

 source of light be revolved about B as a centre through the angle 

 ABC, so that the inclination of BS to BC may become equal to the 

 former inclination of BS to BA. The illumhiation of the line BC, or 

 of any part of it, as BF, will now be equal to the former illumination 

 of BA. But BF and BA are both intercepted between the parallels 

 AX, BT'; hence the quantity of light emanating from BF is equal to 

 that formerly emanating from BA. The revolution of BS about B 

 has increased the angle SBT' by the amount ABC; but SBT' is the 

 supplement of the magnitude of the phase, and ABC is equal to the 

 angle of elevation of the serrations. Accordingly, when the phase of 

 the smooth cylinder is less than the phase of the furrowed cylinder by 

 an amount equal to this angle of elevation, equal quantities of light 

 will reach the observer from the surfaces of which BF and BA are 

 respectively the sections. 



If the position of tlie observer is changed, so that the angle CBT' 

 becomes equal to the angle of elevation ABC or ACB, the point F 

 coincides with C ; in this case, equal quantities of light are received 

 from BC in the smaller, and from BA in the greater phase. For still 

 smaller values of CBT', we have merely to substitute for BA that 

 part of it still visible to the observer. Hence, while the phase remains 

 so small that no pair of adjacent slopes, like AB and AC, are at once 



